Scattering of an attractive Bose-Einstein condensate from a barrier: Formation of quantum superposition states

Scattering in one dimension of an attractive ultracold bosonic cloud from a barrier can lead to the formation of two nonoverlapping clouds. Once formed, the clouds travel with constant velocity, in general different in magnitude from that of the incoming cloud, and do not disperse. The phenomenon and its mechanism—transformation of kinetic energy to internal energy of the scattered cloud—are obtained by solving the time-dependent many-boson Schrödinger equation. The analysis of the wave function shows that the object formed corresponds to a quantum superposition state of two distinct wave packets traveling through real space.

Figure 1

(Color online). Scattering of a solitonic wave packet initially located at $x=0$ and moving with a constant velocity from Gaussian barriers of different widths $\sigma$ centered at $x=-3$. Shown is the density as a function of time. Left panel: full transmission case. Right panel: full reflection case. Middle panel: formation of a superposition state of two distinct wave packets traveling through real space with constant velocity smaller in magnitude from that of the incoming cloud. See text for more details. All quantities are dimensionless.

Figure 2

(Color online). Natural occupation numbers ${\rho }_i\left(t\right)$ in % during the scattering processes of Fig. 1. At $t=0$ all initial wave packets are the same and condensed: ${\rho }_1\left(0\right)=99.1\%$ and ${\rho }_2\left(0\right)=0.9\%$. Until $t=3$ the systems propagate without dispersion, reflecting the solitonic character of the initial wave packet. In the full reflection $\left(\sigma =0.10\right)$ and transmission $\left(\sigma =0.20\right)$ cases the systems remain mainly condensed all the time. In the split case $\left(\sigma =0.15\right)$, after interaction with the barrier, the system evolves into a twofold fragmented state with essentially time-independent occupation numbers ${\rho }_1=59.5\%$ and ${\rho }_2=40.5\%$. All quantities are dimensionless.

Figure 3

(Color online). Proof that the split object corresponds to a superposition state of two distinct wave packets. Left panel: evolution of the expansion coefficients in Fock space spanned by the $|N,0⟩,|N-1,1⟩,\dots ,|1,N-1⟩,|0,N⟩$ configurations. The probabilities ${\left|C_n\left(t\right)\right|}^2$ are plotted as a function of time. The initial wave packet is described essentially by $|N,0⟩$. Right upper and lower panels: natural orbitals ${\left|{\varphi }_i\left(x,t\right)\right|}^2$ (upper two curves of each of the right panels, in green and blue; normalized to 1) and densities (lower curve of each of the right panels, in red) at $t=0$ and $t=20$. The barrier is also shown (indicated in black and localized around $x=-3$). Mainly the $|N,0⟩$ and $|0,N⟩$ configurations contribute to the split object. All quantities are dimensionless.